IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 26, NO. 11, NOVEMBER 2007 1515 Probabilistic Inference on Q-ball Imaging Data Hubert M. J. Fonteijn*, Frans A. J. Verstraten, and David G. Norris Abstract—Diffusion-weighted magnetic resonance imaging (MRI) and especially diffusion tensor imaging (DTI) have proven to be useful for the characterization of the microstructure of brain white matter structures in vivo. However, DTI suffers from a number of limitations in characterizing more complex situations. The most notable problem occurs when multiple fibre bundles are present within a voxel. In this paper, we have expanded the existing Q-ball imaging method to a Bayesian framework in order to fully characterize the uncertainty around the fibre directions, given the quality of the data. We have done this by using a recently proposed spherical harmonics decomposition of the diffusion-weighted signal and the resulting Q-ball orientation distribution function. Moreover, we have incorporated a model selection procedure which determines the appropriate smoothness of the orientation distribution function from the data. We show by simulation that our framework can indeed characterize the posterior probability of the fibre directions in cases with multiple fibre populations per voxel and have provided examples of the algorithm’s performance on real data where this situation is known to occur. Index Terms—Bayesian analysis, crossing fibres, diffusion imaging, Q-ball imaging. I. INTRODUCTION O VER the last two decades, diffusion magnetic resonance imaging (MRI) and especially diffusion tensor imaging (DTI) [1] has proven to be a unique tool for investigating the mi- crostructure of neuronal tissue in vivo. In DTI, the anisotropic behavior of diffusion in white matter is studied. This behavior is caused by the hindrance imposed by axonal membranes on diffusion, which is hypothesized to make diffusion slower per- pendicular to an axonal fibre bundle than parallel to it. This di- rectional information, which is available at the voxel level, has been used to follow tentative fibre tracks through white matter in order to study the anatomical connectivity between brain re- gions [2], [3]. Furthermore, anisotropy indices, which summa- rize the directional variance of the diffusion coefficient, have been used to study differences between patient populations and healthy controls in various diseases that have been associated with white matter abnormalities [4]–[6]. Recently, investigations have been made into the uncer- tainty of all these measures by employing Bootstrap methods Manuscript received May 15, 2007; revised August 21, 2007. This work was supported by a Pionier Grant from the Netherlands Organization for Scientific Research (NWO). Asterisk indicates corresponding author. *H. M. J. Fonteijn is with the Helmholtz Institute, Universiteit Utrecht, 3584 CS Utrecht, The Netherlands and with the F. C. Donders Centre for Cognitive Neuroimaging, 6500 HB Nijmegen, The Netherlands (e-mail: h.m.j.fonteijn@uu.nl). F. A. J. Verstraten is with the Helmholtz Institute, Universiteit Utrecht, 3584 CS Utrecht, The Netherlands (e-mail: f.a.j.verstraten@uu.nl). D. G. Norris with the F. C. Donders Centre for Cognitive Neuroimaging, 6500 HB Nijmegen, The Netherlands (e-mail: david.norris@fcdonders.ru.nl). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMI.2007.907297 [7]–[10] or the Bayesian estimation framework [11]–[13]. These methods have, for example, enabled researchers to quantify the uncertainty in the orientation of the principal eigenvector of diffusion. Other authors have used the shape of the diffusion tensor itself as a measure of uncertainty for the fibre direction [14]–[16]. In these papers, it was hypothesized that very anisotropic tensors represent situations in which the uncertainty around the fibre direction is very low and that very isotropic tensors represent the opposite situation. The resulting measure of uncertainty about the fibre direction on the voxel level can be used to perform probabilistic trac- tography. In contrast with deterministic tractography methods, probabilistic tractography methods not only give the most likely connecting pathway, but provide all possible pathways with their associated probability. It has been long recognized that the DTI model is limited in its capacity to characterize the complexity of white matter struc- ture within a voxel. That is, DTI-based analyses are in principle not able to resolve more than one directionally distinguishable fibre population per voxel, which has proven to be a problem throughout a wide variety of brain structures [17]. This problem self-evidently carries over to any probabilistic analysis based on DTI. To avoid these problems, the DTI model has been extended to encompass multiple compartments, all with their own diffu- sion tensor. When the exchange between these compartments is slow, the diffusion-weighted signal can be hypothesized to be a linear summation of the signals of all the separate compart- ments. This model has been used in a deterministic way [18] and in a Bayesian framework [12], [13]. However, Tuch et al. [18] have shown that the extension of this model beyond two com- partments becomes problematic. This is probably due to the fact that when moving towards more than one compartment, the es- timation procedure becomes nonlinear, which requires the use of iterative optimizers. This increases the likelihood of getting trapped in a local minimum of the optimization function and hence not finding the “true” parameter values. Q-ball imaging [19], [20] is a method with a linear estima- tion procedure which was proposed to resolve multiple fibre populations within a voxel. It, hence, does not suffer from the problems of the multitensor models and it avoids the model se- lection procedure which has to be performed to determine the number of tensors to be modeled. It does, however, introduce another model selection problem, in which the investigator has to choose the smoothness with which to reconstruct the func- tion which indicates the fibre orientations. In this paper, we will tackle this last model selection problem and we will implement and investigate a Bayesian framework to characterize the uncer- tainty in the directions of the underlying fibres in a similar way to approaches previously applied to (multiple-) tensor models. To this end, we have utilized a recent reframing of the original 0278-0062/$25.00 © 2007 IEEE